Bounds on Multiple Self-avoiding Polygons

Hong, Kyungpyo ; Oh, Seungsang

Canadian mathematical bulletin, Tome 61 (2018) no. 3, pp. 518-530

Résumé

A self-avoiding polygon is a lattice polygon consisting of a closed self-avoiding walk on a square lattice. Surprisingly little is known rigorously about the enumeration of self-avoiding polygons, although there are numerous conjectures that are believed to be true and strongly supported by numerical simulations. As an analogous problemto this study, we considermultiple self-avoiding polygons in a confined region as a model for multiple ring polymers in physics. We find rigorous lower and upper bounds for the number ${{p}_{m\times n}}$ of distinct multiple self-avoiding polygons in the $m\,\times \,n$ rectangular grid on the square lattice. For $m\,=\,2,\,{{p}_{2\times n}}\,=\,{{2}^{n-1}}\,-1$ . And for integers $m,\,n\,\ge \,3$ , $${{2}^{m+n-3}}\left( \frac{17}{10} \right){{\,}^{\left( m-2 \right)\left( n-2 \right)}}\,\le \,{{p}_{m\times n}}\,\le \,{{2}^{m+n-3}}\left( \frac{31}{16} \right){{\,}^{\left( m-2 \right)\left( n-2 \right)}}.$$

Détail

DOI : 10.4153/CMB-2017-072-x

Mots-clés : 57M25, 82B20, 82B41, 82D60, ring polymer, self-avoiding polygon